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G = C2×C22⋊C4  order 32 = 25

Direct product of C2 and C22⋊C4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C2×C22⋊C4, C24.C2, C23⋊2C4, C22.12D4, C22.4C23, C23.6C22, C2.1(C2×D4), C22⋊2(C2×C4), (C22×C4)⋊1C2, (C2×C4)⋊3C22, C2.1(C22×C4), SmallGroup(32,22)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2 — C2×C22⋊C4
Chief series
C1 — C2 — C22 — C23 — C24 — C2×C22⋊C4
Lower central
C1 — C2 — C2×C22⋊C4
Upper central
C1 — C23 — C2×C22⋊C4
Jennings
C1 — C22 — C2×C22⋊C4

Generators and relations for C2×C22⋊C4
 G = < a,b,c,d | a2=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Subgroups: 94 in 66 conjugacy classes, 38 normal (6 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C22⋊C4, C22×C4, C24, C2×C22⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4

Character table of C2×C22⋊C4

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H
 size 11111111222222222222
ρ111111111111111111111    trivial
ρ21111-1-1-1-11-11-1-11-11-11-11    linear of order 2
ρ3111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ41111-1-1-1-11-11-11-11-11-11-1    linear of order 2
ρ51111-1-1-1-1-11-111-1-11-111-1    linear of order 2
ρ611111111-1-1-1-1-1-11111-1-1    linear of order 2
ρ711111111-1-1-1-111-1-1-1-111    linear of order 2
ρ81111-1-1-1-1-11-11-111-11-1-11    linear of order 2
ρ91-1-111-11-1-1-111ii-i-iii-i-i    linear of order 4
ρ101-1-11-11-11-111-1-iii-i-iii-i    linear of order 4
ρ111-1-11-11-11-111-1i-i-iii-i-ii    linear of order 4
ρ121-1-111-11-1-1-111-i-iii-i-iii    linear of order 4
ρ131-1-11-11-111-1-11i-ii-i-ii-ii    linear of order 4
ρ141-1-111-11-111-1-1-i-i-i-iiiii    linear of order 4
ρ151-1-111-11-111-1-1iiii-i-i-i-i    linear of order 4
ρ161-1-11-11-111-1-11-ii-iii-ii-i    linear of order 4
ρ172-22-222-2-2000000000000    orthogonal lifted from D4
ρ1822-2-22-2-22000000000000    orthogonal lifted from D4
ρ192-22-2-2-222000000000000    orthogonal lifted from D4
ρ2022-2-2-222-2000000000000    orthogonal lifted from D4

Permutation representations of C2×C22⋊C4
►On 16 points - transitive group 16T21
Generators in S16
(1 9)(2 10)(3 11)(4 12)(5 16)(6 13)(7 14)(8 15)

(2 14)(4 16)(5 12)(7 10)

(1 13)(2 14)(3 15)(4 16)(5 12)(6 9)(7 10)(8 11)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
 
G:=sub<Sym(16)| (1,9)(2,10)(3,11)(4,12)(5,16)(6,13)(7,14)(8,15), (2,14)(4,16)(5,12)(7,10), (1,13)(2,14)(3,15)(4,16)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)>;
 
G:=Group( (1,9)(2,10)(3,11)(4,12)(5,16)(6,13)(7,14)(8,15), (2,14)(4,16)(5,12)(7,10), (1,13)(2,14)(3,15)(4,16)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );
 
G=PermutationGroup([[(1,9),(2,10),(3,11),(4,12),(5,16),(6,13),(7,14),(8,15)], [(2,14),(4,16),(5,12),(7,10)], [(1,13),(2,14),(3,15),(4,16),(5,12),(6,9),(7,10),(8,11)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)]])
 
G:=TransitiveGroup(16,21);
 

C2×C22⋊C4 is a maximal subgroup of
 C23⋊C8  C23.9D4  C24⋊3C4  C23.7Q8  C23.34D4  C23.8Q8  C23.23D4  C24.C22  C24.3C22  C23⋊2D4  C23⋊Q8  C23.10D4  C23.Q8  C23.11D4  C23.4Q8  C2×C4×D4  C22.11C24  C23⋊3D4  C22.32C24  C23⋊2Q8  D4⋊5D4  C22.45C24
C2×C22⋊C4 is a maximal quotient of
 C24⋊3C4  C23.7Q8  C23.34D4  C23.23D4  C24.3C22  C23.67C23  C24.4C4  (C22×C8)⋊C2  C23.C23  M4(2).8C22  C23.24D4  C23.36D4  C23.37D4  C23.38D4  C42⋊C22

Matrix representation of C2×C22⋊C4 ►in GL4(𝔽5) generated by

1000
0400
0010
0001
,
1000
0400
0010
0004
,
1000
0100
0040
0004
,
2000
0400
0004
0040
G:=sub<GL(4,GF(5))| [1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[2,0,0,0,0,4,0,0,0,0,0,4,0,0,4,0] >;
 

C2×C22⋊C4 in GAP, Magma, Sage, TeX 

C_2\times C_2^2\rtimes C_4
 
% in TeX
 
G:=Group("C2xC2^2:C4");
 
// GroupNames label
 
G:=SmallGroup(32,22);
 
// by ID
 
G=gap.SmallGroup(32,22);
 
# by ID
 
G:=PCGroup([5,-2,2,2,-2,2,80,101]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^2=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
 
// generators/relations
 

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Character table of C2×C22⋊C4 in TeX

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